Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

2020 
In this paper it is shown that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. Further, for any $t\geq 2$, it is shown that a pair of orthogonal partial Latin squares of order $n$ can be embedded in a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. A consequence of the constructions is that, if $N(n)$ denotes the size of the largest set of MOLS of order $n$, then $N(n^2)\geq N(n)+2$. In particular, it follows that $N(576)\ge 9$, improving the previously known lower bound $N(576)\ge 8$.
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